Image processing of multiphase images obtained via X-ray
microtomography: A review
Schlüter, S., A. Sheppard, K. Brown, and D. Wildenschild (2014), Image
processing of multiphase images obtained via X-ray microtomography: A review,
Water Resources Research, 50, 3615–3639. doi:10.1002/2014WR015256
10.1002/2014WR015256
American Geophysical Union
Version of Record
http://cdss.library.oregonstate.edu/sa-termsofuse
PUBLICATIONS
Water Resources Research
REVIEW ARTICLE
10.1002/2014WR015256
Key Points:
First survey of image processing
methods for multiphase fluid images
A novel protocol is suitable for
various types of porous media
Many routines come with a freely
available open-source library
Correspondence to:
S. Schl€
uter,
schluets@onid.orst.edu
Citation:
Schl€
uter, S., A. Sheppard, K. Brown, and
D. Wildenschild (2014), Image
processing of multiphase images
obtained via X-ray microtomography:
A review, Water Resour. Res., 50, 3615–
3639, doi:10.1002/2014WR015256.
Received 4 JAN 2014
Accepted 2 APR 2014
Accepted article online 7 APR 2014
Published online 25 APR 2014
Image processing of multiphase images obtained via X-ray
microtomography: A review
Steffen Schl€
uter1,2,3, Adrian Sheppard2, Kendra Brown1, and Dorthe Wildenschild1
1
School of Chemical, Biological and Environmental Engineering, Oregon State University, Corvallis, Oregon, USA,
Department of Applied Mathematics, Research School of Physics and Engineering, Australian National University,
Canberra, Australian Capital Territory, Australia, 3Department of Soil Physics, Helmholtz-Centre for Environmental
Research—UFZ, Halle, Germany
2
Abstract Easier access to X-ray microtomography (lCT) facilities has provided much new insight from
high-resolution imaging for various problems in porous media research. Pore space analysis with respect to
functional properties usually requires segmentation of the intensity data into different classes. Image segmentation is a nontrivial problem that may have a profound impact on all subsequent image analyses. This
review deals with two issues that are neglected in most of the recent studies on image segmentation: (i)
focus on multiclass segmentation and (ii) detailed descriptions as to why a specific method may fail
together with strategies for preventing the failure by applying suitable image enhancement prior to segmentation. In this way, the presented algorithms become very robust and are less prone to operator bias.
Three different test images are examined: a synthetic image with ground-truth information, a synchrotron
image of precision beads with three different fluids residing in the pore space, and a lCT image of a soil
sample containing macropores, rocks, organic matter, and the soil matrix. Image blur is identified as the
major cause for poor segmentation results. Other impairments of the raw data like noise, ring artifacts, and
intensity variation can be removed with current image enhancement methods. Bayesian Markov random
field segmentation, watershed segmentation, and converging active contours are well suited for multiclass
segmentation, yet with different success to correct for partial volume effects and conserve small image features simultaneously.
1. Introduction
The last decade has seen a tremendous progress in X-ray tomography and imaging techniques providing
new means to analyze a multitude of research problems in porous media research. In the scope of water
resources research, applications range from soil-water-root interactions and mechanical and hydraulical
properties of rocks to pore-scale modeling of multiphase flow and continue to appear in related fields of
research (see reviews by Blunt et al. [2013], Cnudde and Boone [2013], Wildenschild and Sheppard [2013], and
Anderson and Hopmans [2013]). Progress in image progressing has kept a comparable pace in terms of new
developments in image enhancement, image analysis, and hardware architectures [e.g., Ketcham and Carlson, 2001; Sheppard et al., 2004; Kaestner et al., 2008; Porter and Wildenschild, 2010; Tuller et al., 2013]. Since
X-ray tomography is becoming a standard technique available to an increasing number of research groups
in water resources research, more and more scientists have a need for information on how to process their
data. Not everyone new to the field has the resources to develop their own image processing toolbox, tailored for the research question at hand, or the budget to take advantage of powerful image processing software that often has a rather comprehensive scope. A relief in this regard are software toolboxes which are
freely available to the scientific community like IMAGEJ [Ferreira and Rasband, 2012], ITK [Ibanez et al., 2005],
QUANTIM [Vogel et al., 2010], BLOB3D [Ketcham, 2005], or SCIKIT-IMAGE [van der Walt et al., 2014], just to name a
few. Their multiphase segmentation capabilities are somewhat limited and may require substantial operator
input. The software used in this study is described in the Appendix A.
However, comparing the performance of different image processing methods on the same set of test
images often leads to very different results. A notorious example is image segmentation of a gray value
image into objects and background [Sezgin and Sankur, 2004; Iassonov et al., 2009; Baveye et al., 2010]. Yet,
these comparative studies often merely list the performance of several segmentation methods with respect
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to a certain quality measure or highlight the user-dependency of the segmentation result, but lack in useful
information as to why a specific method fails under certain circumstances and how this may be avoided by
suitable preprocessing. Another shortcoming is that many recent review papers on image segmentation
with a focus on soil images deal with binary segmentation only [Baveye et al., 2010; Wang et al., 2011; Houston et al., 2013a] and do not provide solutions to multiclass segmentation problems.
This review paper has two main objectives. First, we survey various segmentation methods with respect to
multiclass segmentation. We focus on methods that operate on a single image, i.e., coupled images
scanned at different X-ray energy levels [Rogasik et al., 1999; Costanza-Robinson et al., 2008; Armstrong et al.,
2012] or in a wet and dry state [Culligan et al., 2004; Wildenschild et al., 2005] are not discussed here. We
refer the reader to Brown et al. [2014] where we demonstrate that a single-energy method outperforms a
three-energy method and discuss the potential shortcomings of either approach. All of the surveyed methods are locally adaptive, i.e., in addition to global histogram information they consider some neighborhood
statistic for class assignment. In particular, we will examine hysteresis segmentation [Vogel and Kretzschmar,
1996], indicator kriging [Oh and Lindquist, 1999], converging active contours [Sheppard et al., 2004], watershed segmentation [Vincent and Soille, 1991], and Bayesian Markov random field segmentation [Kulkarni
et al., 2012]. As quality measures, we will use misclassification error, volume fraction, specific interfacial area,
and a connectivity measure. Second, we point out that the performance of image segmentation cannot be
examined independently of image enhancement prior to classification. To do so, we compare the impact of
different denoising methods on the segmentation results. We have chosen standard noise removal methods that were reviewed for application in lCT data of porous media before [Kaestner et al., 2008; Tuller et al.,
2013]. Moreover, we apply efficient algorithms for image artifact removal, such as intensity bias [Iassonov
and Tuller, 2010] and ring artifacts [Sijbers and Postnov, 2004]. Finally, we illustrate the impairment of
proper threshold detection that is due to low contrast and imbalanced histograms, and present methods to
correct it.
The performance of different segmentation methods is evaluated by means of three test images. We start
with a synthetic test image of a partially saturated packing of spheres, where the volume fractions and
interfacial areas of the wetting, nonwetting, and solid phase are known exactly. The true image is superimposed with ring artifacts, blur, and noise, and the success of different combinations of denoising and segmentation in recovering the morphological properties of the true image is compared. Subsequently, the
most suitable combinations are applied to two real images of quite different scopes. The first is a synchrotron image of a three-fluid medium impaired by intensity variation and noise [Brown et al., 2014]; the second is a mCT image of a soil with macropores, organic matter, and rocks impaired by noise and blur
[Houston et al., 2013b].
The paper is organized as follows: in section 2, we provide the details for each image processing method,
while image enhancement and segmentation results are compared in terms of visual appearance and morphology measures in section 3. In section 4, we discuss the results and provide recommendations for best
practices, and our findings are summarized in section 5.
2. Methods
2.1. Artifact Removal
Due to shortcomings in the image acquisition process, the base signal of an X-ray scan is often superimposed by different kinds of image artifacts [e.g., Ketcham and Carlson, 2001; Wildenschild et al., 2002]. The
most frequent impairments are image noise due to a low count of incoming radiation at the detector, and
image blur due to movement, hardware constraints, or suboptimal image reconstruction. As discussed
below, there are powerful denoising methods that efficiently remove noise in homogeneous locations and
at the same time conserve edges between objects. Other image artifacts which are less trivial to remove a
posteriori are ring artifacts, due to defective diodes in the detector panel, or beam hardening of polychromatic beams, which manifests itself in the reconstructed image as streakings around high-attenuation
objects and intensity variation with distance to the sample center. Note that there are means to avoid some
of these artifacts already during image acquisition or image reconstruction like a slightly altering detector
panel position during scanning and wedge calibration [Ketcham and Carlson, 2001]. Here we focus on methods that can be directly applied to the reconstructed volume.
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Ring artifacts can be removed separately for each image slice z after transforming the image from Cartesian
coordinates x 5 (x, y) into polar coordinates x5ðr; uÞ, where x is the location vector, (x, y) is the horizontal
and vertical coordinate, and ðr; uÞ are radius and angle. In this way, the rings appear as vertical lines that
can be removed with a moving window W of width w R, where R is the radius of the sample. The window
detects average variations in gray value median ~IðrÞ along r and normalizes them subsequently [Sijbers and
Postnov, 2004]. Only homogeneous rows within W contribute to the median ~IðrÞ at column r, where the
homogeneity threshold H has to be set by the user. This method has some shortcomings since objects
aligned to a certain radius may also be removed.
To our knowledge, the removal of streaking artifacts due to beam hardening is an unresolved problem.
Intensity bias, on the other hand, can be removed rather easily given that it is not superimposed by changing attenuation coefficients due to variable fluid saturation or matrix porosity. To do so, requires an iterative
procedure [Iassonov and Tuller, 2010]: (i) the image is segmented into the class with the highest attenuation
and background by simple, histogram-based thresholding. (ii) The mean gray value within the highest density class is stored as a function of radius. (iii) A smooth function is fitted to the data:
IðrÞ5a1b cos ð2pr=RÞ1c exp ðr=RÞ
(1)
where a, b, and c are fitting parameters. (iv) The smooth function is used to normalize the data. Steps (i–iv)
are repeated until convergence is achieved after two to four iterations.
2.2. Denoising
2.2.1. Median Filter
A good noise removal algorithm should exert significant smoothing in homogeneous regions (i.e., zones
with low-intensity gradient rI(x)) and minimal modification of edges (i.e., high rI(x) zones), where r is the
@
@
@
differential operator with respect to three dimensions r5 @x
1 @y
1 @z
. The simplest method for nonlinear
denoising is a median filter (MD) with a cubic kernel of diameter d:
^I MD ðxÞ5I0 ðxÞ Md ðxÞ
(2)
where * denotes convolution, I0 is the raw image, and ^I is the denoised result. The gray value that divides
the set of d3 sorted gray values within Md into equal halves is assigned to the current voxel at location x
[Gonzalez and Woods, 2002]. Note that this routine is usually applied in one loop and is rather slow for large
kernel sizes, mainly due to sorting. However, a tremendous increase in speed is achieved by applying
lookup tables and a moving median, i.e., for a kernel shift of one position only a small amount of d2 gray values has to be replaced in a table [Huang et al., 1979].
2.2.2. Anisotropic Diffusion Filter
Another popular, nonlinear denoising method is the anisotropic diffusion (AD) filter [Perona and Malik,
1990; Catte et al., 1992]. The rationale of this method is that the Gaussian distribution is the solution to the
diffusion equation with a constant diffusion coefficient D. In the same way, applying the diffusion equation
with a nonlinear diffusion coefficient amounts to smoothing with a Gaussian kernel of strongly varying
standard deviation. Obviously, D should depend on the local intensity gradient rI(x). Hence, anisotropic diffusion calls for a numerical solution of the following partial differential equation (PDE):
^I 0 5I0
(3)
@^I AD
5r½DðjrðGr ^I AD ÞjÞr^I AD @t
(4)
where t is numerical time, ^I AD is short for ^I AD ðx; tÞ, and the gradient of smoothed intensity values, convolved by a Gaussian Gr of standard deviation r, serve as an edge detector. The simplest implementation is:
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(
DðjrðGr ^IÞjÞ5
1;
jrðGr ^IÞj j
0;
jrðGr ^IÞj > j
(5)
where j is a diffusion stop criterion. The number of iterations is another important parameter that has to be
set manually, because the solution would eventually converge to uniform intensity.
2.2.3. Total Variation Filter
Another PDE-based approach is total variation (TV) denoising [Rudin et al., 1992]. The rationale behind
this method is to minimize the intensity variation in the image by means of the following cost
function:
9
>
>
>
ð
=
2
0
^I TV 5 argmin
jr^IðxÞjdx 1 k jI ðxÞ2^IðxÞj dx
>
>
^I
>
>
>
;
:|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}>
8
>
>
>
<ð
regularization
(6)
fidelity
where k is a scale parameter that controls the trade-off between regularization, i.e., smoothing, and fidelity
to the raw data I0 . The solution is achieved with the following set of coupled PDE’s:
^I 0 5I0
!
@^I TV
r^I TV
5r
1k I0 2^I TV 1K
@t
jr^I TV j
@K
5aðI0 2^I TV Þ
@t
(7)
(8)
(9)
where ^I TV is short for ^I TV ðx; tÞ. The time step control a can be made adaptive to @K/@t. The number of iterations, used as a stopping criterion, is less crucial as compared to ^I AD , because the solution does not converge to uniform intensity due to the fidelity term used in equation (6).
2.2.4. Nonlocal Means Filter
Unlike the previous methods, the nonlocal means filter (NL) is a linear filter, i.e., the gray value at the current
location is the average of gray values at other locations, assigned with some suitable weighting factors, w.
However, in contrast to standard linear filters (Gaussian filter, mean filter, etc.), it does not use a small-sized
kernel, but potentially the entire image as a search window. The rationale is to compare the neighborhoods
of all voxels y 僆 I with the neighbors of the current voxel at location x [Buades et al., 2005]:
^I NL ðxÞ5
X
wðx; yÞI0 ðyÞ
(10)
y2I
Thus, the similarity of a whole neighborhood with fixed size determines the weight w(x, y) with which a distant voxel will influence the new value of the current voxel. More specifically, the weights are significant
only if a Gaussian kernel Gr with standard deviation r around y looks like the corresponding Gaussian kernel around x:
0
ð
1
2
0
0
ðnÞ
jI
ðx1nÞ2I
ðy1nÞj
dn
G
r
B
C
1
C
wðx; yÞ5
exp B
@2
A
h2
ZðxÞ
(11)
where n scans the neighborhood, h acts as a filtering parameter that can be adapted to the level of image
noise and Z(x) is the normalizing factor. Note that the computational cost for the neighborhood search in
the entire image can become excessive, so restricting the search to a certain window size (y 僆 S in equation
(10)) is required [Buades et al., 2008].
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2.3. Edge Enhancement
A notorious problem in image processing is partial volume effects due to image blur. That is, image edges
do not manifest themselves as crisp intensity steps, but rather as gradual intensity changes spanning several voxels. A standard method to sharpen the image, i.e., to enhance the intensity gradient locally, is
unsharp masking [Sheppard et al., 2004]:
0
^I UM ðxÞ5I0 ðxÞ2 wðGr I ðxÞÞ
12w
(12)
where r should roughly match the half width of blurry edges and w defines the degree of edge enhancement, where [0.1, 0.9] is a suitable range. In the context of the last section, this corresponds to the inverse
diffusion equation. Evidently, unsharp masking will also enhance noise, so the image should be denoised
first. Alternative edge enhancement methods like a difference of Gaussians or a Laplacian of Gaussian filter
[Gonzalez and Woods, 2002; Russ, 2006] have a very similar concept and are not further discussed here.
2.4. Image Segmentation
2.4.1. Histogram Bias Correction
The frequency distribution of gray values can have an unfavorable shape for threshold detection. Typical
examples are low contrast, imbalanced class proportions, class skewnesses, or class variances. Here we list
three methods that can partially remove these histogram traits and thus facilitate more reliable threshold
estimates:
1. Gradient mask:
Partial volume effects due to blurred phase edges can cause long tailings in the histogram, which lead to
skewed distributions for the lowest and highest intensity class. Partial volume voxels can be identified
through high-gradient intensities and treated by different strategies [Panda and Rosenfeld, 1978]. One is to
calculate the average gray value of partial volume voxels and use it as an optimal threshold [Schl€
uter et al.,
2010], another is to mask them out and only calculate the histogram for low-gradient regions. The mask is
generated by unimodal thresholding [Rosin, 2001] of the histogram of intensity gradients.
2. Histogram equilization:
Image contrast is often enhanced by linear or nonlinear intensity rescaling, sometimes also denoted as histogram stretching [Gonzalez and Woods, 2002; Russ, 2006]. An alternative approach to contrast enhancement is contrast-limited adaptive histogram equalization (CLAHE) [Pizer et al., 1987]. Image contrast can be
defined as the slope of the cumulative density function of gray values. Limiting this contrast corresponds to
clipping the histogram at a certain cutoff. The histogram area thus removed is uniformly distributed over
the range of gray values that occur in the image. In principle, this algorithm operates on the histogram of a
certain search window to obtain a locally adaptive contrast enhancement. The method can be generalized
such that any transition between global and local contrast enhancement is achieved [Stark, 2000]. In this
study, histogram clipping is only applied to the global histogram to improve threshold detection and is not
mapped to the corresponding image.
3. ROI dilations:
Some images exhibit unimodal histograms due to very imbalanced class proportions, i.e., a very small volume fraction of a certain phase and a very large volume fraction of the background. A balanced frequency
distribution can be obtained with a new, semiautomatic algorithm: (i) pick a threshold manually that detects
the class with lowest volume fraction as the region of interest (ROI) and binarize the image. (ii) Dilate the
thus obtained mask. (iii) Multiply the original image with the mask in order to compute the ROI histogram.
(iv) If it is not yet clearly bimodal (multimodal) return to step (ii).
2.4.2. Global Thresholding
Image segmentation is a crucial step in image processing and affects all subsequent image analyses. In this
context, it is common to refer to global thresholding as approaches where classes are assigned to voxels by
histogram evaluation only, without considering how the gray values are spatially arranged in the corresponding image. A multitude of different thresholding methods exist today which have been reviewed by
various authors [Sahoo et al., 1988; Pal and Pal, 1993; Trier and Jain, 1995; Sezgin and Sankur, 2004]. The
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general conclusion, if any, is that none of the reviewed methods excel at all segmentation problems. A comprehensive survey by Sezgin and Sankur [2004] compared 40 different thresholding methods, most of them
global, and classified them according to fundamental principles. Five out of those methods were chosen for
implementation in this study as well as an additional one:
1. G1—Maximum Variance:
This is a classic method based on discriminant analysis [Otsu, 1975]. Consider the histogram as an estimator
of the probability of a certain gray value and n – 1 thresholds to divide the histogram into n classes
{C0, C1,. . .,Cn}. The total variance r2T of the population of gray values can be divided into the sum of withinclass variances r2W and the between-class variance r2B of class means. The objective is to find the set of n – 1
thresholds ft0 ; t1 ; . . . ; tn21 g that maximize this between-class variance.
2. G2—Minimum Error:
Minimum Error Thresholding assumes the histogram to be composed of normal distributions for each class
[Kittler and Illingworth, 1986]. The Gaussian modes that are fitted to the histogram usually overlap at certain
gray values. As a consequence, assigning those voxels to only one class will deliberately lead to a certain
misclassification error for the other class. The objective is to set the thresholds ft0 ; t1 ; . . . ; tn21 g such that
the misclassification error is minimal.
3. G3—Maximum Entropy:
This method exists in various modifications. The classic approach, which is implemented here, relies on
Shannon Entropy as a measure of the information content of a signal [Kapur et al., 1985]. Assume a
threshold to be close to the maximum or minimum gray value. Objects will barely appear in the output
image and will be almost completely surrounded by background. Hence, the information content of the
resulting image is low. A set of thresholds ft0 ; t1 ; . . . ; tn21 g can be adjusted such that the image has the
richest detail, i.e., the information transfer is optimal. This is achieved by maximizing the sum of class
entropies.
4. G4—Fuzzy C-means:
This method combines the classic k-means algorithm [Ridler and Calvard, 1978] with fuzzy set theory [Jawahar et al., 1997]. Membership functions M0, M1, . . . Mn are assigned to each gray value depending on the
distance to each class mean l0, l1, . . . ln and a fuzziness index s. The optimal set of thresholds
ft0 ; t1 ; . . . ; tn21 g is detected at the intersections of adjacent M.
5. G5—Shape:
This method detects the thresholds ft0 ; t1 ; . . . ; tn21 g at the local minimum between two adjacent histogram
peaks [Tsai, 1995]. If the number of peaks np exceeds the predefined number of classes n, iterative Gaussian
smoothing is first applied to the histogram, until np 5 n. If np < n, the missing thresholds are set at the location of maximum histogram curvature instead.
6. G6—Average:
It can be shown that some methods are optimal under certain conditions [Kurita et al., 1992], e.g.,
equal class probabilities, equal class variances, etc. However, these conditions are hardly ever met in a
real image and all methods will be biased to some degree. Assuming that the bias of different methods may partly cancel out due to the different criteria that they optimize, an averaged threshold over
all methods may lie closer to the true, unknown optimum. Since some methods may fail completely,
outliers have to be removed, i.e., only thresholds within ðt k 2rtk ; t k 1rtk Þ contribute to the final value,
where k 5 0,. . ., n – 1.
Note that in the first three methods the computational cost for an exhaustive search over all sets of thresholds in a n-dimensional search space increases exponentially with increasing number of classes n. Therefore,
the methods are implemented in an efficient way by using lookup tables to store each term of a specific
objective function for every possible pair of class boundaries, and employ these tables to calculate the
objective function for an arbitrary number of classes [Liao et al., 2001]. For the fuzzy c-means, the exhaustive
search is replaced by an iterative search [Jawahar et al., 1997]. The search space for the shape method is
one-dimensional irrespective of the number of classes.
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2.4.3. Local Segmentation
In contrast to global, histogram-based thresholding, locally adaptive segmentation methods also account
for some kind of neighborhood statistic for class assignment in order to smooth object boundaries, avoid
noise objects, or compensate for local intensity changes. Due to the added flexibility, local segmentation
methods often result in more satisfying segmentation results [Iassonov et al., 2009; Wang et al., 2011]. Five
different local segmentation methods, which have all been successfully applied to porous media images in
the past, will be used in this study:
1. L1—Hysteresis:
Hysteresis segmentation, which is sometimes also denoted as bilevel segmentation or region growing, was
introduced as a method to improve edge continuity in gradient images [Canny, 1986]. In the same way, it
improves the class assignment of partial volume voxels in soil images [Vogel and Kretzschmar, 1996; Schl€
uter
et al., 2010]. Two thresholds have to be set by the user, a lower threshold that identifies voxels, which definitely belong to a low-intensity class, and an upper threshold for which the uncertainty of class assignment
is highest. A voxel in the intermediate gray value range is only assigned to the low-intensity class, if a neighbor voxel already belongs to the low-intensity class. In other words, low-intensity voxels serve as seed
regions for iterations of conditional dilations. Unassigned voxels which cannot be accessed by this region
growing process are assigned to the high-intensity class. Hysteresis thresholding is not a multiclass segmentation method in its strictest sense as the procedure has to be repeated n – 1 times for n classes.
2. L2—Indicator kriging:
In this geostatistical method, spatial correlation is used as a local assignment criterion [Oh and Lindquist,
1999; Houston et al., 2013b]. Again, two thresholds are specified by the user to define two a priori classes.
The upper threshold of the unclassified range is extended toward gray values that definitely belong to a
high-intensity class. The class assignment at an unclassified location depends on the weighted average of
indicator values in its neighborhood, where the weights are obtained by kriging. Indicator kriging has to be
repeated n – 1 times for n classes as well.
3. L3—Bayesian Markov Random Field:
^
The rationale of this probabilistic segmentation method is to find a spatial arrangement of class labels C
with minimum boundary surface that at the same time honors the gray value data in the best possible way
[Berthod et al., 1996; Kulkarni et al., 2012]. This is a combinatorial optimization problem which is solved in
the framework of Markov random fields (MRF), i.e., by only evaluating the interaction between direct
neighbors:
9
8
>
>
>
>
>
>
ð
ð
qffiffiffiffiffiffiffiffiffiffi 0
=
<
I
2l
c
x
^ argmin
1
b
cðc
;
c
Þ
2pr2c 1
C5
x y
2
>
>
2rc
^
X
P
>
>
C
>
;
:|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} >
(13)
class boundaries
class statistics
with
(
cðcx ; cy Þ5
21;
cx 5cy
11;
cx 6¼ cy
(14)
where X is the population of all voxels, P is the population of all pairs of neighboring voxels x and y, cx is
the class label at x, lc and r2c are class mean and variance, and b is a homogeneity parameter that determines the weight of the penalty term for class boundaries. Class updating is achieved in a deterministic
order denoted with iterative conditional modes (ICM) [Besag, 1986], where sufficient convergence is usually
achieved after three to five loops.
4. L4—Watershed:
The watershed algorithm uses lines of highest gradient to demarcate class borders locally [Beucher and Lantuejoul, 1979; Vincent and Soille, 1991; Roerdink and Meijster, 2000]. The image is preclassified with simple
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thresholding to set markers for the immersion process. High-gradient zones are subsequently set to unclassified again, as they will be assigned by the watershed algorithm. These high-gradient zones are either identified by edge detection (Sobel, Canny, etc.) or by a small cubic kernel that detects neighborhoods with
nonuniform class assignment. The latter method produced the better results for the images examined in
this study. The unclassified zones are filled from different ends using the intensity gradient as a pseudoheight until the watershed line is reached. Finally, the voxels located directly on the separation line and on
plateaus are filled with the most representative class in their neighborhood.
5. L5—Converging Active Contours:
This approach is a combination of the watershed method and active contour methods, which uses gradient
and intensity information simultaneously [Sheppard et al., 2004]. The method is initialized by identifying
seed regions for each class; as in the methods above, this means that each voxel is either assigned to one
of the classes or left unassigned, to be classified during the main segmentation step. Like the watershed
method described above, this initial classification is normally done by thresholding in both intensity and
gradient space; each class has an upper and a lower threshold along with an upper gradient threshold. The
main classification algorithm proceeds by simultaneously growing the boundaries of these seed regions
toward each other. The speed at which the boundaries advance varies spatially and temporally, depending
on the local gradient and optionally on the distance of the local gray value to its class mean. The algorithm
ends when all boundaries have converged. The final separation line assignment problem described above
that affects some watershed methods is not an issue because every nonseed voxel is traversed by one class
boundary before any others. The advancement of the boundaries is implemented efficiently using the fast
marching algorithm; despite this, the method is quite computationally intensive and requires parallel implementations to operate on large 3-D images.
2.5. Postprocessing
Denoising and boundary refinement can also be applied to the class image C as postprocessing instead of
smoothing the raw image I0 . Among the most popular postprocessing methods for binary images are morphological operators like erosion and dilation [Serra, 1982]. However, in multiclass images they would correspond to a minimum and maximum filter, which are not suited for this purpose at all. A median filter, on
the other hand, removes segmentation noise and rugged class boundaries rather well. It does however,
have the unwanted feature that the results happen to depend on the order in which the classes are num^ MA , which assigns the most representative class among
bered. This is avoided by applying a majority filter C
all neighbors in a cubic kernel to the central voxel. It is common to enforce two criteria to prevent
unwanted features where more than two phases meet: (i) the most representative class exceeds a certain
volume fraction and (ii) the volume fraction of the most representative class exceeds the volume fraction of
the old class at the central voxel by a certain percentage.
An alternative postprocessing method is size-dependent object removal. For this method, each object of
each class has to be labeled. Volumes are then determined by voxel counting, and objects smaller than a
user-dependent threshold are filled with the class that completely surrounds the object. Consequently,
rugged surfaces of big objects are conserved. Small deleted objects at the boundary between two materials
are filled by simultaneous dilation of both materials.
2.6. Structural Analysis
The list of tools with which complex structures like porous media can be analyzed is virtually endless. This
study is constrained to only four very simple, but meaningful metrics. If ground-truth information is available from a true image I, a misclassification error ME can be determined:
Nx
1 X
ME5
dðci ; ^c i Þ;
Nx i51
(
dðci ; ^c i Þ5
1;
ci 6¼ ^c i
0;
ci 5^c i
(15)
where ci and ^c i are the true and current class at location i and Nx is the number of all voxels. Bulk volumes V
and surface areas a are examined for each individual phase. Specific interfacial areas between two phases
(a and b) are obtained by the following relation:
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aab 5 ðaa 1ab 2ac Þ
2
(16)
where c is the union of all other classes. Both V and a are determined with Minkowski functionals [Vogel
et al., 2010]. In this way, surface area estimates are directly obtained from the segmented voxel image and
problems associated with aligning isosurfaces of each individual phase are avoided. Finally, a dimensionless
connectivity indicator for a specific phase is calculated [Renard and Allard, 2013]:
Ca 5
Nl
1 X
n2
Na2 i51 i
(17)
where each cluster of phase a has a label li and a size ni, Nl is the number of clusters and Nn is the number
of all phase a voxels. Cluster labeling is typically achieved with a fast method by Hoshen and Kopelman
[1976]. Ca has the edge over the popular Euler number that it is bounded by (0, 1] and less sensitive to
noise.
3. Results
3.1. Synthetic Test Image
3.1.1. Image Generation
Synthetic images provide ground-truth data against which different image enhancement and segmentation
methods can be compared. The test image in this study is generated to resemble a partially saturated sand
packing. To this end, a volume of 512 3 512 3 128 voxels is filled by nonoverlapping spheres with a radius
of r 5 50 voxels placed at random locations. If a new sphere does not fit at a given location, the radius
decreases in steps until the overlap vanishes. The location is completely abandoned when r 5 20 is reached.
The procedure stops after 250 spheres have successfully been placed resulting in a porosity of 0.412. Subsequently, an opening transform [Serra, 1982] is applied to the pore space with a spherical structure element
of r 5 15. The pore space which has been removed by this opening is considered to be filled with the wetting phase (w) and the remaining, larger pore bodies are assigned to the nonwetting phase (n). This does
not necessarily produce a physically realistic fluid distribution, but evokes fluid configurations that are similar enough to serve as a test scenario. The resulting volume fractions are Vw 5 0.279 and Vn 5 0.133. Gray
values of 50, 125, and 205 are assigned to nonwetting, wetting and solid phase, respectively, resulting in
the true image I. An axial image slice of I is depicted in Figure 1a.
Subsequently, the image is superimposed with ring artifacts, blur, and noise in order to obtain a more realistic, raw image I0 (Figure 1b). The ring artifacts are additive with random height and random radius. The rings
manifest themselves as a random gray value offset in the range [2100,100], where the resulting gray value
is limited at [0,255]. Subsequently, blurring is achieved with a cubic mean kernel of five voxels on a side.
Finally, uncorrelated Gaussian noise is added with a signal-to-noise ratio of rb/rn 5 2, where rb and rn are
the standard deviations of gray values in the blurred image and the noise model, respectively.
3.1.2. Denoising and Edge Enhancement
Four different denoising methods (Figures 1d–1g) are applied to the raw image I0 in Figure 1b in order to
remove noise followed by an edge enhancement step to mitigate partial volume effects caused by blur.
The median filter was applied one time with a cubic convolution kernel with a side length of d 5 7 voxels.
The anisotropic diffusion filter was applied with j 5 15, r 5 1.0, and t 5 1.5 (nine iterations). The total variation filter was applied with k 5 2 stopping after t 5 0.12 (30 iterations). For the nonlocal means filter, a cubic
kernel of d 5 9 was used for neighborhood search with a Gaussian convolution kernel of r 5 2 in a reduced
search window S of d 5 23 voxels. The two criteria for the majority filter are set to an absolute majority of
0.33 and a higher occurrence than the class of the central voxel of 0.05.
The settings of each denoising method have been carefully chosen to achieve strong denoising, deliberately accepting some degree of edge smoothing. Subsequently, an unsharp mask filter for edge enhancement with a Gaussian kernel of r 5 1.5 and a weighting factor of w 5 0.7 was applied to each denoising
result. Finally, the results of the unsharp mask after anisotropic diffusion ^I AD1UM and total variation
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frequency
c
(b)
(d)
(e)
(f)
(g)
0.08
0.06
0.05
0.04
0.03
0.02
0.01
0
0
d
e
25
50
f
75
100 125 150
grey value
175
200
225
250
g
Figure 1. (a) True image I with spheres and size-dependent distribution of wetting and nonwetting phase, (b) raw image I0 exhibiting ring artifacts, blur and noise, (c) histograms before
and after image enhancement. (bottom row) Image enhancement results with different denoising methods in combination with unsharp masking: (d) ^I MD1UM , (e) ^I AD1UM , (f) ^I TV1UM , and
(g) ^I NL1UM .
denoising ^I TV1UM had to be cleaned by a median filter with a d 5 5 kernel to smooth rough surfaces that
were enhanced during unsharp masking.
The histograms in Figure 1c and visual appearances in Figures 1d–1g indicate different success in noise
removal and edge preservation. The edges in the median image ^I ME1UM appear slightly more blurry than in
the images for the other three denoising methods. Moreover, we observe that the nonlocal means filter
^I NL1UM is superior in removing ring artifacts during the denoising step alone and does not require the additional cleanup of rough edges by a median filter mentioned above.
3.1.3. Ring Artifact Removal
The ring artifact routine of Sijbers and Postnov [2004] is applied slice wise with a window size of w 5 27 pixels and a homogeneity threshold H adapted to the noise level in Figures 1d–1g. The outcome is depicted
for the enhanced image ^I TV1UM after TV denoising and unsharp mask edge enhancement (Figure 2c). Obviously, the ring artifact removal works well for some rings, but is incomplete for others even after carefully
testing different w and H. For instance, the white ring within the green window exhibits a varying magnitude after denoising (Figure 2a). Since the high artifact magnitude in the pore space contributes less than
a
b
c
Figure 2. (a) Test image after a combination of TV denoising and unsharp masking ^I TV1UM (corresponds to Figure 1f). Yellow frame marks
sharp boundaries before ring artifact removal and the green ring corresponds to the window in polar coordinates. (b) Same image in polar
coordinates after line removal. Green rectangle depicts the moving window W. (c) Same image after ring artifact removal. The yellow frame
highlights blur due to the back transform into Cartesian coordinates.
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75
0.93
tl0-th0
l h
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tmax
[0,1]
50
25
0.92
0.91
0.9
b
INL+UM
ground truth
tmax
[0,1]
0.01
0.001
0.0001
25
50
100
150
200
t0 - threshold dark-to-medium
0.07
50
75
100
125
150
gray value
175
200
225
0.012
tl[0,1]-th[0,1]
INL+UM
G1
G2
G3
G4
G5
G6
0.05
0.04
0.03
c
d
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frequency
0.06
frequency
tl0-th0
l h
t1-t1
0.1
frequency
a
B
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t1 - threshold medium-to-bright
225
σ2(t0,t1) / ⟨σ2(t0,t1)⟩max - between-class var.
Water Resources Research
0.008
0.006
0.004
0.02
0.002
0.01
0
0
25
50
75
100
125 150
gray value
175
200
225
25
50
75
100
125
150
gray value
175
200
225
Figure 3. (a) normalized objective function of Otsu’s method (G1) with the thresholds detected at the maximum between-class variance and the transition regions stopping at the 0.992
percentile. (b) Histogram of ^I NL1UM after ring artifact removal with thresholds t0max and t1max and transition ranges (½t0l ; t0h and ½t1l ; t1h ) detected with Otsu’s method (G1). (c) Threshold
l
; th ½0; 1 are obtained accordingly. (d) Same averaging
pairs for five different methods (G1–G5) and the corresponding averages after outlier removal (G6). The transition regions ½t½0;1
method after edge masking and histogram clipping.
half to the window height in Figure 2b, it remains undetected by the median ~IðrÞ. Stretching the window
over more than one slice as suggested by Ketcham [2006] does not improve the results. In addition, the
back transform from polar into Cartesian coordinates introduces additional blur for high radii (yellow
frame).
3.1.4. Global Thresholding
In Figure 3a, the well-known Otsu method (G1) is applied as an example to the histogram of ^I NL1UM after
ring artifact removal. Results for the other denoising methods are similar (not shown). The exhaustive search
for each pair of thresholds detects the optimum at tmax 5 {95; 166}. Percentiles of this objective criterion
can be used to obtain fuzzy threshold ranges [Oh and Lindquist, 1999], i.e., t1 is held constant and t0
decreases (or increases) until in this case r2b =hr2b imax 50:992 is reached at t0l (or t0h ). The procedure is
repeated for t1 where t0 kept is constant. Applying these thresholds and associated ranges to the underlying
histogram (Figure 3b) reveals a bias of Otsu’s method toward the class with highest volume fraction. Note
that the true optimum, which is estimated by the intersection points of the individual gray value frequencies within in each class of the true image, is at t true 5f91; 162g. Some other thresholding methods are also
afflicted by bias, yet in different directions (Figure 3c). Fuzzy c-means (G3) show the same bias due to imbalanced class probabilities. The shape analysis (G5) performs well since the local histogram minima happen to
coincide with the intersection points in Figure 3b. The minimum error method (G2) is biased due to imbalanced skewness, i.e., only the wetting phase exhibits two long tailings due to partial volume effects in both
directions, which increases the probability of being assigned to that class. The maximum entropy method
(G4) fails completely in identifying meaningful thresholds. Figure 3c also depicts the arithmetic mean of
thresholds and associated ranges after outlier removal (G6). This average set of thresholds at t 5 {88; 165} is
closer to the true optimum. Note that the specific percentiles for each thresholding method have to be set
individually, as each objective criterion exhibits very different ranges, i.e., some are bounded and yet others
are logarithmic. A large part of the bias in Figure 3c can be removed easily. First, the impact of imbalanced
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nonwetting
wetting
solid
d
Figure 4. Segmentation results for different local segmentation methods applied on the image after nonlocal means denoising and
unsharp masking ^I NL1UM : (a) original image, (b) global thresholding, (c) indicator kriging, (d) Bayesian MRF, (e) watershed, and (f) converging active contours. Differently colored frames highlight failures of various methods.
skewness is mitigated by masking out partial volume voxels at phase edges [Panda and Rosenfeld, 1978].
Second, the impact of imbalanced class probabilities is reduced by histogram clipping [Pizer et al., 1987]. In
this way, the skewness of the nonwetting and solid phase is also efficiently removed. The combination of
both methods leads to a well-balanced histogram for which each of the surveyed thresholding methods
ends up at similar values (Figure 3d). The average after outlier removal at t 5 {90; 162} matches the true
optimum almost perfectly.
3.1.5. Local Segmentation
The average set of thresholds (Figure 3d) are determined for each denoising method individually and then
used for global thresholding as an initial step in all local segmentation methods. Note that a priori segmentation with Bayesian Markov random field (L3) and watershed segmentation (L4) requires single thresholds
tkmax only, hysteresis segmentation (L1) uses the range ½tkl ; tkmax , and indicator kriging (L2) and converging
active contours (L5) use ½tkl ; tkh , with k 5 {0; 1}. In addition, converging active contours requires a gradient
threshold, which is determined automatically by unimodal thresholding of the gradient histogram.
The results of each local segmentation method on ^I NL1UM are depicted in Figure 4. The segmentation result
for hysteresis segmentation (L1) is left out, because it is indistinguishable from the outcome of global thresholding (G6). All segmentation results are free of ring artifacts and noise and exhibit smooth object boundaries
which can be mainly credited to successful preprocessing. However, some segmentation results perform better than others in the way they recover image features impaired by blur. Incorrectly identified, apparent wetting films that are due to partial volume voxels at class boundaries between the nonwetting and solid phases
(yellow frame) can be mainly attributed to image blur. Indicator kriging (L2) and hysteresis segmentation (L1)
cannot cope with this problem, since these are iterative methods, i.e., the segmentation between wetting and
nonwetting phase is independent of the segmentation between wetting and solid phase.
In principle, Bayesian MRF segmentation (L3) has a mechanism to remove these films. To do so, the homogeneity factor b has to be set high, thus also removing true features of similar size. A moderate value of
b 5 0.5 is used here. Yet, no matter how high b is set only direct neighbors are evaluated in the MRF paradigm and since the films are thicker than two voxels they cannot be penalized. Another fundamental shortcoming of Bayesian MRF segmentation that has previously gone unnoticed is its tendency toward bias in
the class statistics (equation (13)). The histogram of the wetting phase exhibits long tailing toward both
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0
40
preprocessed
thresholds
80
120
160
gray value
non-wetting
wetting
200
240
modified wet.
solid
Figure 5. First term of the objective function for MRF segmentation (equation (13)) as a function of gray value for each class. In the modified class statistics, rw is replaced by rw/2 for the wetting phase.
directions in contrast to only one tail for the other classes (Figure 3c). Consequently, rw is much higher than
rn and the probability of a voxel in the intermediate intensity range to be assigned to the wetting phase
increases accordingly, resulting in even thicker apparent wetting films. This bias is mitigated here by a quick
and rather inelegant fix to use rw/2 for the class statistics instead. In this way, the gray values at which the
penalty functions for adjacent classes cross each other coincide much better with the previously detected
thresholds (Figure 5).
The watershed method (L4) successfully removes apparent wetting films in the yellow frame (Figure 4e).
However, true wetting films are also being removed (white frame). Segmentation with converging active
contours (L5—Figure 4f) exhibit excessive wetting film removal as well. Finally, no method is capable of
restoring the small wetting phase bridge (green frame) in Figure 4, because its intensity is highly smoothed
due to image blur.
3.1.6. Structural Analysis
The shortcomings of the different segmentation methods are corroborated by structural properties of the
nonwetting phase listed in Table 1. The misclassification error ME is roughly 3–4% for all methods including
global thresholding. However, the lowest ME does not guarantee the best recovery of morphological properties. The bulk volume Vn of the nonwetting phase is slightly underestimated by all segmentation methods,
except for watershed segmentation and converging active contours which show a tendency for overestimation, due to the way partial volume voxels are treated. Surprisingly, simple thresholding and hysteresis
thresholding match the true an value best, whereas indicator kriging and Bayesian MRF underestimate it
and watershed segmentation and converging active contours overestimate it as a consequence of partial
volume voxel treatment. More importantly, the specific interface between fluids awn only diverges from the
total nonwetting surface an if false wetting films are successfully suppressed. Simple thresholding, hysteresis
thresholding, indicator kriging, and Bayesian MRF segmentation all fail in this respect. The only exception is
^ MA , simply because the
Bayesian MRF segmentation on the raw data if combined with postprocessing C
majority filter removes partial volume voxels. Watershed segmentation and converging active contours
underestimate awn because a lot of true wetting films are removed as well. The connectivity index of the
true image Cn is matched well by most segmentation methods, because the bubbles remain rather isolated
no matter how wetting films are treated. Oversegmentation of the nonwetting phase with watershed segmentation and converging active contours may lead to lower Cn, however, because (i) the denominator Na
in equation (17) increases and (ii) the increase in bubble volume is evenly distributed among all clusters in
the numerator. The differences in structural properties due to different denoising methods are also listed in
Table 1. Obviously, no denoising at all produced the worst results in all respects. Moreover, the structural
properties do not vary much among the different denoising methods. Surprisingly, a simple majority filter
^ MA , applied on the segmented raw data without any preprocessing results in the best agreement with the
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Table 1. Structural Properties of Segmented Images for Different Combinations of Denoising and Segmentation Methods: Unsharp
Mask (UM), Median (MD), Anisotropic Diffusion (AD), Total Variation (TV), Nonlocal Means (NL), Majority (MA)a
ME (–)
Vn (–)
an (pix21)
anw (pix21)
Cn (–)
0.000
0.133
0.0169
0.0140
0.355
0.140
0.028
0.029
0.027
0.027
0.024
0.141
0.128
0.124
0.127
0.130
0.129
0.1005
0.0168
0.0165
0.0169
0.0172
0.0169
0.0954
0.0167
0.0165
0.0169
0.0170
0.0168
0.312
0.355
0.355
0.355
0.354
0.355
0.107
0.027
0.029
0.027
0.027
0.024
0.131
0.127
0.123
0.127
0.129
0.128
0.0722
0.0167
0.0164
0.0169
0.0171
0.0167
0.0695
0.0167
0.0164
0.0169
0.0170
0.0167
0.346
0.355
0.355
0.355
0.354
0.355
0.037
0.033
0.037
0.032
0.033
0.029
0.131
0.128
0.122
0.125
0.127
0.131
0.0202
0.0166
0.0162
0.0165
0.0166
0.0167
0.0201
0.0166
0.0162
0.0165
0.0166
0.0167
0.386
0.351
0.353
0.352
0.352
0.352
0.049
0.027
0.033
0.028
0.028
0.038
0.133
0.124
0.119
0.124
0.127
0.134
0.0282
0.0163
0.0160
0.0165
0.0166
0.0170
0.0272
0.0163
0.0160
0.0165
0.0165
0.0140
0.385
0.356
0.357
0.356
0.356
0.356
0.041
0.029
0.025
0.029
0.032
0.037
L5—Converging Active Contours
0.046
I0
^I MD1UM
0.042
^I AD1UM
0.042
^I TV1UM
0.042
^I NL1UM
0.043
^ MA
0.043
C
0.136
0.135
0.134
0.135
0.137
0.136
0.0184
0.0172
0.0170
0.0174
0.0175
0.0174
0.0112
0.0110
0.0117
0.0111
0.0102
0.0101
0.358
0.355
0.355
0.355
0.322
0.359
0.136
0.136
0.137
0.136
0.138
0.136
0.0241
0.0173
0.0172
0.0174
0.0175
0.0175
0.0178
0.0108
0.0107
0.0110
0.0108
0.0105
0.390
0.324
0.324
0.324
0.324
0.360
Denoising
True Image
G6—Global Thresholding
I0
^I MD1UM
^I AD1UM
^I TV1UM
^I NL1UM
^ MA
C
L1—Hysteresis
I0
^I MD1UM
^I AD1UM
^I TV1UM
^I NL1UM
^ MA
C
L2—Indicator Kriging
I0
^I MD1UM
^I AD1UM
^I TV1UM
^I NL1UM
^ MA
C
L3—Bayesian MRF (b 5 0.5)
I0
^I MD1UM
^I AD1UM
^I TV1UM
^I NL1UM
^ MA
C
L4—Watershed
I0
^I MD1UM
^I AD1UM
^I TV1UM
^I NL1UM
^ MA
C
a
ME is misclassification error, Vn is volume fraction of nonwetting phase, An is surface area density of nonwetting phase, Anw is surface
area density between nonwetting and wetting phase, and Cn is connectivity index of nonwetting phase.
true bulk properties. This is because any kind of preprocessing, i.e., denoising and edge enhancement,
removes structural information to some degree.
3.1.7. Final Workflow
At this point, some preliminary conclusions about a suitable image processing protocol can already be
drawn in order to set the workflow for the remaining images:
1. The variability in structural properties among the different denoising methods was rather small. Qualitatively, all filters reduce noise in homogeneous areas well. In addition, only the nonlocal means filter sufficiently denoises along edges while smoothing across the edges is inhibited. In this way, an additional
treatment of rugged surfaces after edge enhancement is not necessary. Thus, in the remainder of the paper,
only the nonlocal means filter will be used.
2. A combination of methods for histogram bias correction causes a better agreement between different
threshold detection methods. The outlier-corrected average of five different threshold detection methods
reproduced ground-truth information well and will be used for global thresholding.
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Figure 6. Workflow diagram for multiclass segmentation of the remaining multifluid image and soil image. Headlines denote image processing steps and gray boxes the specific methods.
3. Hysteresis thresholding and indicator kriging are not capable of multiclass segmentation, which leads to
misclassifaction errors where more than two phases meet locally. Therefore, only converging active contours, watershed and Bayesian MRF segmentation will be applied to the remaining images. Simple thresholding will also be applied for sake of comparison.
4. Segmentation of the raw data with a subsequent majority filter performed slightly better in terms of structural
properties than denoising the intensity data in advance. Both approaches will be compared with each other.
These findings translate into the workflow diagram shown in Figure 6. Some steps are optional, e.g., the
need for edge enhancement depends on the sharpness of the raw image, ROI dilations are only necessary,
if the volume fraction of the phase of interest is very low, etc.
3.2. Synchrotron Image of Three-Fluid Phases in a Porous Medium
3.2.1. Image Enhancement
A sample of sintered glass beads with a porosity of roughly 32% has been scanned at a resolution of 9.24
lm. The pore space was partially saturated with air (14%), oil (39%), and water (47%). The cylindrical region
of interest has a diameter of 665 voxels and height of 210 voxels. The image is free of ring artifacts and is
sharp, without noticeable blur (Figure 7a). The reconstruction method caused a slight decrease in mean
intensity for large radii which is evident in the denoised image after nonlocal means denoising (Figure 7b).
This radial intensity variation can be removed almost completely (Figure 7c) with intensity bias correction
[Iassonov and Tuller, 2010]. This has a considerable impact on the histogram (Figure 7d). The frequency distributions for each class turns from broad and multipeaked into a narrow band. For the same reasons as discussed for the synthetic test image, edges are masked out and the histogram is clipped into a wellbalanced frequency distribution, so that five different global threshold detection methods (G1–G5) yield
very similar sets of thresholds. Again, the average after outlier removal (G6) is used for final segmentation
using local methods. Note that edge enhancement with unsharp mask filtering has not been applied, as
there are hardly any partial volume voxels in this image.
3.2.2. Image Segmentation
Simple thresholding (G6), Bayesian MRF segmentation (L3), watershed segmentation (L4), and converging
active contours (L5) are applied to the multifluid image, either on the preprocessed ^I NL or on the raw I0 followed by a majority filter. A small subset of the segmentation results are depicted in Figure 8. Since the
exact arrangement of interfaces is unknown, it is difficult to judge objectively what combination of methods
performs best. The edge between beads and air is roughly two voxels thick. In the smooth ^I NL image, the
edge is assigned to oil films, water films or both (Figures 8d, 8f, 8h, and 8j), whereas in the noisy image
edge voxels are equally assigned to all four classes but subsequently assigned to air or beads since they
constitute the most representative class in the neighborhood of an edge (Figures 8c, 8e, 8g, and 8i). It is
almost impossible to conclusively determine whether a fluid film thicker than the image resolution really
covers the beads entirely. However, small isolated water voxels between solid voxels and oil film are highly
unlikely, and should be suppressed. Global thresholding without postprocessing has no mechanism to
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b
0.045
0.035
c
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(a)
(b)
(c)
0.04
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e
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G2
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0.02
0.015
G3
0.005
G4
0.004
G5
0.003
G6
0.002
0.01
0.001
0.005
0
0
0
50
100
150
200
250
25
50
75
100
gray value
125
150
175
200
225
gray value
Figure 7. Image enhancement of multifluid image: (a) raw image I0 , (b) nonlocal means filter ^I NL on I0 , (c) beam hardening removal on ^I NL . The yellow frame outlines the subset in Figure 8.
(d) Histograms of Figures 7a–7c. (e) Global threshold detection after histogram clipping.
achieve this. Bayesian MRF segmentation of ^I NL only succeeded in doing so with a relatively high homogeneity parameter (b 5 10). In turn, watershed segmentation fails to detect an oil film at the left side of the
pendular water ring, either because of a lacking seed voxel for oil or because the underlying gradient image
is not sharp enough to evoke two distinct edges within such a short distance. Converging active contours
result in subjectively plausible results for both I0 and ^I NL .
3.2.3. Image Analysis
The qualitative analysis illustrated in Figure 8 is corroborated by the results with respect to structural analysis in Table 2. The bulk volumes of air and oil, Va and Vo, remain largely unaffected by the choice of
global
a
MRF
c
watershed
CAC
e
g
i
f
h
j
raw
solid
oil
air
NL means
water
b
d
Figure 8. Segmentation results for multifluid image: (a) raw image I0 , (b) nonlocal means filter ^I NL , global thresholding on (c) I0 with postprocessing or (d) on ^I NL , Bayesian MRF segmentation (e) with b 5 0.1 on I0 with postprocessing or (f) with b 5 10 on ^I NL , watershed segmentation (g) on I0 with postprocessing or (h) on ^I NL , converging active contours (i) on I0 with postprocessing or (j) on ^I NL .
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Table 2. Volume Fractions V and Surface Area Densities a of Air (a) and Oil (o) for the Segmented Multifluid Imagea
Denoising
Va (–)
aa (mm21)
Vo (–)
ao (mm21)
aao (mm21)
G6—Global Thresholding
^ MA
C
^I NL
0.047
0.046
0.400
0.545
0.125
0.125
1.621
2.281
0.120
0.384
0.047
0.046
0.046
0.046
0.482
0.505
0.459
0.397
0.125
0.124
0.125
0.125
1.840
2.214
2.063
1.686
0.215
0.369
0.327
0.149
0.048
0.048
L5—Converging Active Contours
^ MA
0.047
C
^I NL
0.048
0.403
0.406
0.127
0.127
1.646
1.631
0.107
0.109
L3—Bayesian MRF
^ MA ðb5 0:1Þ
C
^I NL ðb5 0:1Þ
^I NL ðb51Þ
^I NL ðb510Þ
L4—Watershed
^ MA
C
^I NL
0.421
0.407
0.121
1.642
0.130
0.123
1.650
0.119
a
^
is either applied prior to segmentation with a nonlocal means filter I NL or as postprocessing with a majority filter
Denoising
^ MA .
C
segmentation methods. The surface area densities ao, aa, and aao, however, vary considerably among different segmentation methods. This is because small image objects like films, meniscii, and ganglia exhibit
large surface-to-volume ratios and at the same time are associated with the highest uncertainty in terms of
^ MA leads to a general reduction in surface areas and to very similar
class assignment. A majority filter C
results for all segmentation methods. The surface areas increase with a decreasing boundary penalty factor
b for Bayesian MRF segmentation on ^I NL and are highest for global thresholding on ^I NL , where there is no
^ MA ,
penalty term at all. The ao and aa values for b 5 10 are very similar to all postprocessed class images C
to the outcome of converging active contours on ^I NL , and to the result after watershed segmentation on
^I NL . Yet, the specific surface area between air and oil (aao 5 0.149) is as much as 27% higher than the
average of other methods (aao 5 [0.107 – 0.130]). Eventually, a decision as to which image is closer to reality
needs to be made. But without an analytical solution or other ground-truth information such a decision will
suffer from a certain level of subjectivity.
3.3. lCT Image of Soil
3.3.1. Image Enhancement
The soil image corresponds to image 2 in Houston et al. [2013b] with a size of 256 3 256 3 256 voxels and
a resolution of 32 lm. The sample consists of macropores (Vp 12%), organic matter (Vo 13%), soil matrix
(Vm 72%), and dense particles like rocks (Vr 3%). The raw data I0 (Figure 9a) is again denoised with a
nonlocal means filter ^I NL (Figure 9b). Afterward, edges are enhanced by an unsharp mask ^I NL1UM with r 5 1
and w 5 0.5 so that intensity values of partial volume voxels are forced closer to their respective class means
(Figure 9c). The gradient mask in Figure 9d detects partial volume voxels at phase edges which are
excluded from subsequent threshold detection. Each of these methods has a favorable impact on the intensity histogram (Figure 9e) in that valleys between the class peaks are much more pronounced. The combination of all methods, i.e., edge masking on ^I NL1UM , together with histogram clipping leads to a modified
histogram for which all five threshold detection methods (G1–G5) lead to similar results (Figure 9e). The
average after outlier removal (G6) is again used for all locally adaptive segmentation methods.
3.3.2. Image Segmentation
Only some of the segmentation methods introduced in this paper are applied to the soil image. Global
^ MA (Figure 10a) shall serve as a refthresholding of the raw data I0 in combination with a majority filter C
erence to which the other methods can be compared. Evidently, simple thresholding already leads to
rather satisfying results if it is accompanied by suitable postprocessing. However, the segmented image
clearly suffers from false organic coatings around macropores which can be attributed to incorrect
assignment of partial volume voxels (violet frame). This misclassification of boundary voxels can be
avoided with watershed segmentation (Figure 10b) and converging active contours (Figure 10c) both
applied to denoised and edge-enhanced image ^I NL1UM . In addition, even thin macropores are correctly
detected (green frame). Bayesian MRF segmentation is applied to ^I NL1UM with b 5 0.1, b 5 1, and b 5 10,
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0.025
a
c
e
a
b
0.02
c
0.015
0.01
frequency
d
clip
0.005
0
0
40
80
120
160
200
240
gray value
d
f
0.01
0.008
0.006
frequency
b
combined
G1
G2
G3
G4
G5
G6
0.004
0.002
0
0
40
80
120
160
200
240
gray value
Figure 9. Image enhancement of a soil image: (a) raw image I0 [Houston et al., 2013b], (b) ^I NL after nonlocal means denoising, (c) ^I NL1UM after unsharp mask, (d) gradient mask on
^I NL1UM , (e) histogram of Figures 9b–9d and histogram clipping of Figures 9b. (f) Histogram after combined postprocessing (b 1 c 1 d 1 clip) together with the corresponding thresholds
obtained by various global thresholding methods.
respectively (Figures 10d–10f). We observe that the segmented images look very different for different b. If
the penalty term for class boundaries (second term of equation (13)) has a low weight (b 5 0.1), the image
looks very similar to (Figure 10a), i.e., all image objects are well preserved, yet all macropores exhibit false
coatings of organic matter. In turn, if the homogeneity parameter is set very high (b 5 10), partial volume
effects are suppressed and so are small image objects in general, like the thin macropore in the green
frame. A moderate value (b 5 1) leads to an unsatisfactory trade-off between the two problems.
3.3.3. Image Analysis
The qualitative interpretation is again corroborated by structural properties of macropores and organic matter
summarized in Table 3. Bulk volume and surface area vary much more among different segmentation methods as compared to the multifluid image. For instance, the surface area density aop between organic matter
and macropores decreases by a factor of 2.5 if the homogeneity parameter for Bayesian MRF segmentation is
increased from b 5 0.1 (Figure 10d) to b 5 10 (Figure 10f), and is even smaller for watershed segmentation
(Figure 10b) and converging active contours (Figure 10c). In addition, the connectivity indicator C is also sensitive to the choice of segmentation method, because small objects close to the image resolution have a high
impact on the continuity of a phase. Therefore, macropore connectivity increases from Cp 5 (0.72 – 0.80) to
Cp 5 (0.83 – 0.84) if thin macropores that connect larger pore bodies are correctly identified, e.g., the green
frame in Figure 10. At the same time, the connectivity of organic matter decreases drastically from Cp 5 (0.94
– 0.96) to Cp 5 (0.64 – 0.74) if false organic coatings of macropores (violet frame) are successfully suppressed.
4. Discussion
4.1. Image Enhancement
We have demonstrated how essential a suitable combination of image enhancement methods can be for
subsequent image segmentation. The correction of intensity bias due to beam hardening was adequately
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a
10.1002/2014WR015256
b
c
e
f
pore
soil matrix
rock
organic
matter
d
Figure 10. Image segmentation of the soil image: (a) global thresholding on I0 with postprocessing, (b) watershed segmentation on ^I NL1UM , (c) converging active contours on ^I NL1UM ,
(d–f) Bayesian MRF segmentation on ^I NL1UM with b 5 {0.1, 1, 10}, respectively.
corrected by the method of Iassonov and Tuller [2010], since the examined sample was cylindrical and fairly
homogeneous. If the intensity bias had a more complex shape, which cannot be readily described as function of radius, the bias model could have been obtained by interpolation instead. For instance, the
approach of Yanowitz and Bruckstein [1989] to use the Laplace equation to interpolate a threshold surface
between slowly varying gray values along phase boundaries can be adapted to interpolate a bias surface
between objects of the highest intensity class. Even better results can be expected by using the Poisson
equation for this purpose [Perez et al., 2003]. The ring artifact removal routine [Sijbers and Postnov, 2004]
succeeded in removing most of the rings. However, the ring artifact removal is incomplete if the artifact
magnitude is not constant along rotation angle /. Moreover, the back transform from polar into Cartesian
Table 3. Volume Fractions V, Surface Area Densities a, and Connectivity Indices C of Pores p and Organic Residues o for the Soil Imagea
Denoising
Vp (–)
G6—Global Thresholding
^ MA
0.082
C
^I NL
0.096
L3—Bayesian MRF
^ MA ðb5 0:1Þ
0.085
C
^I NL ðb5 0:1Þ
0.093
^I NL ðb51Þ
0.086
^I NL ðb510Þ
0.087
L4—Watershed
^ MA
0.110
C
^I NL
0.117
L5—Converging Active Contours
^ MA
0.088
C
^I NL
0.129
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Vo (–)
ao (mm21)
aop (mm21)
Cp (–)
Co (–)
0.213
0.190
2.74
3.30
0.91
1.29
0.721
0.777
0.953
0.940
0.208
0.197
0.193
0.176
2.63
2.97
2.13
1.46
0.91
1.12
0.71
0.44
0.734
0.787
0.795
0.822
0.953
0.950
0.961
0.963
0.137
0.127
1.05
0.84
0.30
0.25
0.802
0.834
0.652
0.644
0.135
0.116
1.24
0.66
0.40
0.24
0.730
0.825
0.744
0.681
a
Denoising
is either applied prior to segmentation with a nonlocal means filter ^I NL or as postprocessing with a majority filter
^ MA .
C
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coordinates introduces additional blur that increases with radius r. Thus, the performance is always somewhat worse as compared to line removal directly applied on the sinograms when this is possible [Ketcham,
2006]. Also, some methods operate on rings in Cartesian space directly [Freundlich, 1987]. Finally, Fourier
and wavelet filters generally lead to improved ring removal [Raven, 1998; M€
unch et al., 2009].
The surveyed denoising methods were efficient in removing noise while keeping the blurring of edges at a
minimum, given that the associated parameters are set adequately. In fact, image noise today has become
a secondary issue for successful image analysis. Instead, image blur has been identified as more of a pitfall
for the success of the various segmentation methods in this study. Edge enhancement with unsharp masks
partly mitigates image blur, but is not capable of removing partial volume effects completely. Surely, future
advances in X-ray tomography hardware and reconstruction software will lead to steady improvements in
image quality, so that sharper images can be acquired on a routine basis.
One of our significant findings was that even though image enhancement is often indispensable for robust
threshold detection, it does not necessarily imply that the segmentation itself also has to be applied to the
enhanced image. Instead, segmenting the raw data with the thus obtained thresholds may lead to fewer
misclassification errors if suitable postprocessing is applied to the segmented images. This is because any
image enhancement inevitably destroys some structural information in the raw data. For the images examined in this study, a majority filter on the segmented raw data produced good results, mostly because the
objects had rather smooth, convex boundaries. In turn, a majority filter can be a less desirable option if true
objects are thin, concave, have rough surfaces or acute angles. It is up to the user to always compare and
decide which is the most favorable option.
4.2. Global Thresholding
We have demonstrated that every histogram-based thresholding method relies on certain assumptions
about the histogram shape. This introduces some bias if the class modes have different variance, skewness,
or proportion. Many bias correction techniques for the standard methods used here have been suggested.
For instance, minimum error thresholding can be corrected for imbalanced overlap [Cho et al., 1989] or can
be used with Poisson distributions for each class instead of Gaussian distributions [Pal and Bhandari, 1993].
Shannon entropy can be replaced by Tsallis entropy for maximum entropy thresholding, allowing for an
additional degree of freedom that can be used to tune the results [de Albuquerque et al., 2004]. Fuzzy cmeans can be corrected for imbalanced class proportions [Jawahar et al., 1997]. The list is virtually endless.
However, we found it more useful to (i) put some effort into suitable image enhancement prior to thresholding, (ii) alleviate the impact of bias by histogram clipping, and (iii) use the average after outlier removal
to determine thresholds for subsequent segmentation. Even with this preprocessing methodology there
might be soil images that still exhibit unimodal histograms due to a lot of unresolved porosity or very
imbalanced class proportions [Wang et al., 2011; Baveye et al., 2010]. The second problem can be avoided
with a semiautomatic algorithm based on ROI dilations (Figure 11). For instance, small rocks constitute only
3% to volume in the soil image and are hard to distinguish as an individual class in the histogram. The
threshold for the region of interest (ROI) is set to the class mean (t 5 ls 5 203). Consecutive dilations of the
ROI mask lead to a clearly bimodal histogram for which a threshold between the soil and rock class can be
easily identified. Note that the thus obtained local histogram minimum at 182 is very close to t 5 185 in Figure 9f, but much easier to identify. An alternative to deal with unimodal histograms, which is, however,
restricted to two-class segmentation, is to estimate a threshold from the mean gray value within edge
regions [Panda and Rosenfeld, 1978; Schl€
uter et al., 2010]. The problem of too much unresolved porosity or
too gradual intensity changes is more severe and puts the entire concept of segmentation into question. In
this case, some morphology analysis can be deployed to the intensity data directly, including distance transforms [Jang and Hong, 2001], isosurfaces [McClure et al., 2007], skeletonization [Chung and Sapiro, 2000], or
tortuosity [Gommes et al., 2009].
4.3. Local Segmentation
We have corroborated the importance of using locally adaptive methods truly capable of multiclass segmentation instead of applying iterations of binarizations [Tuller et al., 2013]. Methods that fulfill this criterion
are Bayesian MRF segmentation [Berthod et al., 1996; Kulkarni et al., 2012], watershed segmentation [Beucher
and Lantuejoul, 1979; Vincent and Soille, 1991; Roerdink and Meijster, 2000], and converging active contours
[Sheppard et al., 2004]. Bayesian MRF segmentation was originally developed for image classification in the
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0.0014
no mask
t=203
1. iter
2. iter
3. iter
4. iter
5. iter
6. iter
0.0012
frequency
0.001
0.0008
0.0006
0.0004
0.0002
0
0
40
80
120
160
gray value
200
240
Figure 11. Iterative algorithm to identify a truely bimodal histogram at high gray values in the soil image.
presence of additive noise and is based on the assumption that individual class modes follow a Gaussian
distribution. This causes failure of the method for denoised images with pronounced partial volume effects
due to image blur, which leads to either one-sided or two-sided tailings in the histogram. As a consequence,
special care has to be taken to correct for this histogram bias. In addition, the results depend heavily on the
homogeneity parameter b. Erroneous assignment of partial volume voxels could only be suppressed with a
high penalty on class boundaries (b 5 10), which at the same time removed true image features of similar
size. A promising improvement of the Bayesian MRF segmentation, especially if applied to fluid images,
would be to replace the unspecified penalty term (equation (14)) with real surface tensions [Knight et al.,
1990; Silverstein and Fort, 2000]. In fact, any noninvasive laboratory technique which provides independent
measurements about a structural property of the same sample can potentially help to condition segmentation parameters. This has been recently demonstrated for three-phase segmentation of limestone via hysteresis thresholding, where the region growing parameters were conditioned by independent porosity
measurements [Mangane et al., 2013].
For the watershed segmentation in its current implementation, a benefit is that it only requires simple
thresholding and a gradient image. However, the lack of additional parameters also hampers flexibility.
Therefore, excessive elimination of wetting films in the synthetic image could not be avoided. In contrast,
the somewhat similar converging active contours method honors local gradient and intensity information
at the same time. A reported drawback of this method is its sensitivity to seed region detection and the
adjustment of parameters for the speed function [Iassonov et al., 2009]. In this paper, we demonstrated
strategies to define the seed regions automatically. In addition, the parameters for the speed function can
be obtained directly by a careful evaluation of the gradient histogram (Figure 12). The frequency of local
gradients is almost always unimodal, and therefore, difficult to threshold with standard methods. Yet, every
unimodal histogram exhibits some characteristic features such as a point of maximum curvature [Tsai,
1995] or the point of maximum distance from an auxiliary line between the histogram mode and the maximum bin [Rosin, 2001] (Figure 12, inset). In addition, some gradient histograms may exhibit secondary
shoulders if the intensity data comprise fairly constant edge heights. Picking one of these points as a gradient threshold would result in a fully objective segmentation method. In this study, we always used the gradient cutoff detection with Rosin’s method. In turn, adjusting this cutoff adds some flexibility to the speed
function in the same way as b scales the penalty term in the Bayesian MRF method. For instance, we
achieved a very good reproduction of the true interfacial area between fluids awn in the synthetic test image
by setting the gradient cutoff to a higher value.
As expected, objects that are close in size to the image resolution were associated with the highest uncertainty in all test images. Five voxels in diameter has been suggested as a rough estimate for an object size
limit for which image analysis is reliable [Lehmann et al., 2006; Vogel et al., 2010]. Features like the thin oil
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0.1
0.1
frequency
0.08
frequency
0.01
0.06
0.04
0.02
0
0.001
50
100
150
gray value
200
Sobel
max. deviation
max. curvature
local minimum
0.0001
0
25
50
75
100
125
150
gray value
175
200
225
250
Figure 12. Gradient histogram computed on a Sobel image of ^I NL1UM for the synthetic image. Each unimodal histogram has a unique
point of maximum curvature and maximum distance from the auxiliary line depicted in the inset. In addition, the shoulder in the histogram produces a local minimum, which can be detected as well.
films and pendular rings at grain contacts in the synchrotron image clearly do not meet this criterion. Brown
et al. [2014] compared surface area densities of smooth fluid interfaces for the same kind of three-fluid samples when improving the image resolution from 10.6 to 5.3 lm and observed an increase in the range of 5–
16%, depending on the specific fluid pair. An improvement in resolution usually comes at the cost of a
smaller field of view as far as industrial scanners are concerned, as well as increased noise unless the scan
time is increased as well. For heterogeneous media, especially, there is a trade-off between a representative
image size and sufficient resolution to capture important details. However, the scale window can also be
extended toward smaller objects by scanning the same sample at different resolutions, even with different
imaging techniques, and merging the images with image registration [Latham et al., 2008], statistical fusion
of images [Mohebi et al., 2009] or scale fusion applied to structure analysis data only [Vogel et al., 2010;
Schl€
uter et al., 2011]. Finally, for this study, we have merely compared the outcome of different local segmentation methods with each other. Of course, several methods can also be combined. For instance, Bayesian MRF segmentation with surface tensions could be used for postprocessing the outcome of converging
active contours. If necessary, postprocessing with a majority filter could also be applied in addition to
denoising prior to segmentation. After all, the choice of a suitable protocol always depends on characteristics of the raw data.
5. Conclusions
We have surveyed recent advances in image enhancement and image segmentation of multiphase X-ray
microtomography data. Image enhancement methods included ring artifact removal, intensity bias correction, edge enhancement, image denoising, and contrast enhancement. Image segmentation methods comprised six global segmentation and five different locally adaptive segmentation methods. Some general
conclusions can be drawn from our findings:
1. Image blur is the major cause of poor segmentation results in this study, since image noise and other
image artifacts could be removed with current image processing methods.
2. A lot of uncertainty in threshold detection can be removed by suitable preprocessing and correction of
histograms for bias in the class statistics prior to thresholding.
3. Bayesian Markov random field segmentation, watershed segmentation and converging active contours
are suited for multiclass segmentation. The converging active contour method has potentially the highest
flexibility to correct for partial volume effects and simultaneously conserve small image features.
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Appendix A
Most image processing steps presented in this study can be done with freely available software. The only exceptions are the nonlocal means filter for denoising and the watershed method for segmentation, which were carR FIRE (http://www.vsg3d.com/avizo/fire). Moreover, the converging active contours method
ried out with AVIZOV
for segmentation was performed with MANGO (http://physics.anu.edu.au/appmaths/capabilities/mango.php).
The conversion between Cartesian and polar coordinates for ring artifact removal was performed with the
freely available Polar Transformer (http://rsbweb.nih.gov/ij/plugins/polar-transformer.html) plug-in for
IMAGEJ. All other image processing steps are either built on or directly implemented in the QUANTIM (http://
www.quantim.ufz.de) open-source image processing library.
Acknowledgments
We thank Alasdair Houston (SIMBIOS,
Abertay, UK) for providing the lCT soil
image and three anonymous reviewers
for their helpful comments. S. Schl€
uter
is grateful to the Alexander-vonHumboldt Foundation for granting a
Feodor-Lynen scholarship. A. Sheppard
acknowledges the support of an
Australian Research Council Future
Fellowship.
€
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ET AL.
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